International Association
for Cryptologic Research

Laconic cryptography focuses on designing two-message protocols that allow secure computation over large datasets while minimizing communication costs. While laconic cryptography protocols achieve asymptotically optimal communication complexity for many tasks, their concrete efficiency is prohibitively expensive, due to heavy use of public-key techniques or non-black-box cryptographic primitives. In this work, we initiate the study of {\it laconic cryptography with preprocessing}, introducing a model that includes an offline phase to generate database-dependent correlations, which are then used in a lightweight online phase. These correlations are conceptually simple, relying on linear-algebraic techniques. This enables us to develop a protocol for private laconic vector oblivious linear evaluation (plvOLE). In such a protocol, the receiver holds a large database $\mathsf{DB}$, and the sender has two messages $v$ and $w$ along with an index $i$. The receiver learns the value $v \cdot \mathsf{DB}_i + w$ without revealing other information. Our protocol, which draws from ideas developed in the context of private information retrieval with preprocessing, serves as the backbone for two applications of interest: laconic private set intersection (lPSI) for large universes and laconic function evaluation for RAM programs (RAM-LFE). Based on our plvOLE protocol, we provide efficient instantiations of these two primitives in the preprocessing model.

We initiate the study of lower bounding the verification time of Succinct Non-interactive ARguments of Knowledge (SNARKs) built in the Polynomial Interactive Oracle Proof + Polynomial Commitment Scheme paradigm. The verification time of these SNARKs is generally dominated by the polynomial commitment scheme, and so we want to understand if polynomial commitment schemes admit lower bounds on the verification time. By recognizing that polynomial commitment schemes are also often built by applying cryptography to some information-theoretic core protocol, we seek to separate this core from the cryptography in a way that meaningfully captures the verification time required by the polynomial commitment scheme verifier. We provide strong evidence that several polynomial commitment schemes have (nearly) \emph{optimal} verifier times. Our evidence comes from connecting polynomial commitment schemes to certain information-theoretic protocols known as communication protocols from the field of communication complexity, a link which we believe to be of independent interest. Through this lens, we model the verifier work in the cryptographic protocols as information (i.e., number of bits) exchanged between parties in the communication protocols, allowing us to leverage lower bounds from communication complexity. These lower bounds give strong evidence that the verifier time in these polynomial commitment schemes must be at least the number of bits exchanged in the communication protocol. We extract the communication protocol cores of three polynomial commitment schemes and lower bound the bits exchanged in these cores. The lower bounds we obtain match (up to poly-logarithmic factors) the best-known (asymptotic) verification times of the polynomial commitment schemes we examine in this work. Specifically, we show that for univariate/multilinear polynomials of size $N = 2^n$: - the communication core of Hyrax PCS (Wahby et al., S&P 2016) requires $\Omega(\sqrt{N}}$ bits to be exchanged; - the communication core of Bulletproofs PCS (Bootle et al., EUROCRYPT 2016; Bünz et al., S&P 2018) requires $\Omega(N)$ bits to be exchanged; and - the communication core of Dory PCS (Lee, TCC 2021) requires $\Omega(\log(N))$ bits to be exchanged. Our results strongly suggest a negative answer to a longstanding open question on whether the Bulletproofs verifier can be made sublinear time.

Polynomial commitment schemes allow a prover to commit to a polynomial and later reveal the evaluation of the polynomial on an arbitrary point along with proof of validity. This object is central in the design of many cryptographic schemes such as zero-knowledge proofs and verifiable secret sharing. In the standard definition, the polynomial is known to the prover whereas the evaluation points are not private. In this paper, we put forward the notion of \emph{private polynomial commitments} that capture additional privacy guarantees, where the evaluation points are hidden from the verifier while the polynomial is hidden from both. We provide concretely efficient constructions that allow simultaneously batch the verification of many evaluations with a small additive overhead. As an application, we design a new concretely efficient multi-party private set-intersection with malicious security and improved asymptotic communication and space complexities. We demonstrate the concrete efficiency of our construction via an implementation. Our scheme can prove $2^{10}$ evaluations of a private polynomial of degree $2^{10}$ in 157s. The proof size is only 169KB and the verification time is 11.8s. Moreover, we also implemented the multi-party private set intersection protocol and scale it to 1000 parties (which has not been shown before). The total running time for $2^{14}$ elements per party is 2,410 seconds. While existing protocols offer better computational complexity, our scheme offers significantly smaller communication and better scalability (in the number of parties) owing to better memory usage.

Zero-knowledge proofs allow a prover to convince a verifier of a statement without revealing anything besides its validity. A major bottleneck in scaling sub-linear zero-knowledge proofs is the high space requirement of the prover, even for NP relations that can be verified in a small space. In this work, we ask whether there exist complexity-preserving (i.e. overhead w.r.t time and space are minimal) succinct zero-knowledge arguments of knowledge with minimal assumptions while making only black-box access to the underlying primitives. We essentially resolve this question up to polylogarithmic factors. Namely, for every NP relation that can be verified in time T and space S, we construct a public-coin zero-knowledge argument system that is black-box based on collision-resistant hash-functions (CRH) where the prover runs in time $\O(T)$ and space $\O(S)$, the verifier runs in time $\O(T/S+S)$ and space $\O(\kappa)$ and the communication is $\O(T/S)$, where $\kappa$ is the statistical security parameter. Using the Fiat-Shamir heuristic, our construction yields the first complexity-preserving ZK-SNARK based on CRH (via a black-box construction). Furthermore, we give evidence that reducing the proof length below $\O(T/S)$ will be hard using existing techniques by arguing the space-complexity of constant-distance error correcting codes.